few_shots: Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001392

8
options: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
problem_activity: Warm-up
The purpose of this warm-up is for students to recall how to calculate an average rate of change from two points. Students will use this skill throughout the lesson and return to the context and data table in the last activity of the lesson.
Launch
Ask students to close their books or devices. Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then select 3–5 students to share something they notice or wonder about the table  and record and display their responses for all to see. If not pointed out, ask students, "Describe how the data is changing as the value of
\(t\)
is increasing." (The values of
\(p\)
are decreasing at a slower rate.) Ask students to reopen their books or devices and complete the activity. Follow with a whole-class discussion.
Student Facing
Let
\(p\)
be the function that gives the cost
\(p(t)\)
, in dollars, of producing 1 watt of solar energy
\(t\)
years after 1977. Here is a table showing the values of
\(p\)
from 1977 to 1987.
| \(t\) | \(p(t)\) |
|-------|----------|
|   0   |    80    |
|   1   |    60    |
|   2   |    45    |
|   3   |  33.75   |
|   4   |  25.31   |
|   5   |  18.98   |
|   6   |  14.24   |
|   7   |  10.68   |
|   8   |   8.01   |
|   9   |   6.01   |
|  10   |   4.51   |
Which expression best represents the average rate of change in solar cost between 1977 and 1987?
\(p(10)-p(0)\)
\(p(10)\)
\(\dfrac{p(10) - p(0)}{10-0}\)
\(\dfrac{p(10)}{p(0)}\)
Student Response
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Anticipated Misconceptions
Some students might confuse finding an average of the values in the table and finding an average rate of change. Help them see that average usually involves one unit, such as average number of cookies. Average rate of change involves comparing how one quantity changes when another quantity changes by 1, such as cost ($) per year.
Activity Synthesis
The goal of this discussion is for students to recall the meaning of average rate of change for a function and how it is calculated. Select students to describe what the value of each expression represents in this context. For example,
\(p(10)-p(0)\)
is the difference in cost for a solar cell between 1977 and 1987 while
\(\dfrac{p(10)}{p(0)}\)
could be used to identify the percent change from 1977 to 1987. An important takeaway for students is that the actual expression for the average rate of change,
\(\dfrac{p(10)-p(0)}{10-0} \approx \text-7.55\)
, tells us that the price decreased by -
$
7.55 each year on average since the expression looks at the total difference in price between the 1977 and 1987 divided by the total number of years that passed.


correct_option_index: 0